SpectralAnalysis, Stability andBifurcations Systems ...morrison/13BIRS_morrison_CHHII.pdf · systems. In this chapter, we describe some mathematical aspects of the CHH continuing

Nonlinear PhysicalSystems

Spectral Analysis, Stability and Bifurcations

Edited byOleg N. Kirillov

Dmitry E. Pelinovsky

Series EditorNoel Challamel

iSfE WILEY

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, aspermitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers ,or in the case of reprographic reproduction in accordance with the terms and licenses issued by theCLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at theundermentioned address:

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Preface .

Chapter 1. SurprisingDavide BIGONI, Diego

1.1. Introduction .

1.2. Buckling in ten

1.3. The effect of cc

1.4. .The Ziegler per

1.5. Conclusions .

1.6. Acknowledgme1.7. Bibliography.

MIX

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN: 978-1-84821-420-0

A@" -uJ Paper fromFSC responsible sources

wwwfscorg FSee C013604

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CRO 4YY

Chapter 2. WKB Soluand Applications . .Jean-Francois BONY, ~

2.1. Introduction .

2.2. Connection of I

2.2.1. A model in

2.2.2. Classical m

2.2.3. Review of ~

2.2.4. The microh2.2.5. The microl.

2.3. Applications to

2.3.1. Spectral pn2.3.2. Resonance

2.4. Acknowledgrm2.5. Bibliography.

x Nonlinear Physical Systems

Chapter 13. Continuum Hamiltonian Hopf Bifurcation II . . . . . . . . . . 283George I. HAGSTROM and Philip J. MORRISON

13.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28413.2. Mathematical aspects of the continuum Hamiltonian

Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28513.2.1. Structural stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28513.2.2. Normal forms and signature . . . . . . . . . . . . . . . . . . . . . . 287

13.3. Application to Vlasov–Poisson . . . . . . . . . . . . . . . . . . . . . . . 28813.3.1. Structural stability in the space Cn(R)∩L1(R) . . . . . . . . . . . . 29213.3.2. Structural stability in W 1,1 . . . . . . . . . . . . . . . . . . . . . . . 29413.3.3. Dynamical accessibility and structural stability . . . . . . . . . . . 296

13.4. Canonical infinite-dimensional case . . . . . . . . . . . . . . . . . . . . 30013.4.1. Negative energy oscillator coupled to a heat bath . . . . . . . . . . 301

13.5. Commentary: degeneracy and nonlinearity . . . . . . . . . . . . . . . . 30313.6. Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 30813.7. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30813.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

Chapter 14. Energy Stability Analysis for a Hybrid Fluid-KineticPlasma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311Philip J. MORRISON, Emanuele TASSI and Cesare TRONCI

14.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31114.2. Stability and the energy-Casimir method . . . . . . . . . . . . . . . . . 31214.3. Planar Hamiltonian hybrid model . . . . . . . . . . . . . . . . . . . . . . 314

14.3.1. Planar hybrid model equations of motion . . . . . . . . . . . . . . . 31414.3.2. Hamiltonian structure . . . . . . . . . . . . . . . . . . . . . . . . . . 31614.3.3. Casimir invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

14.4. Energy-Casimir stability analysis . . . . . . . . . . . . . . . . . . . . . . 31814.4.1. Equilibrium variational principle . . . . . . . . . . . . . . . . . . . 31914.4.2. Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

14.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32314.6. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32414.7. Appendix A: derivation of hybrid Hamiltonian structure . . . . . . . . . 32414.8. Appendix B: Casimir verification . . . . . . . . . . . . . . . . . . . . . . 32614.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Chapter 15. Accurate Estimates for the Exponential Decay of Semigroupswith Non-Self-Adjoint Generators . . . . . . . . . . . . . . . . . . . . . . . . 331Francis NIER

15.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33115.2. Relevant quantities for sectorial operators . . . . . . . . . . . . . . . . . 334

Chapter 13

Continuum Hamiltonian Hopf Bifurcation II

Building on the development of Chapter 12, bifurcation of unstable modes thatemerge from continuous spectra in a class of infinite-dimensional noncanonicalHamiltonian systems is investigated. A bifurcation called the continuum HamiltonianHopf (CHH) bifurcation is main significance, which is an infinite-dimensional analogof the usual Hamiltonian Hopf (HH) bifurcation. Necessary notions pertaining tospectra, structural stability, signature of the continuous spectra and normal forms aredescribed. The theory developed is applicable to a wide class of 2+1 noncanonicalHamiltonian matter models, but the specific example of the Vlasov-Poisson systemlinearized about homogeneous (spatially independent) equilibria is discussed indetail. For this example, structural (in)stability is established in an appropriatefunctional analytic setting, and two kinds of bifurcations are considered, one atinfinite and one at finite wave number. After defining and describing the notion ofdynamical accessibility, Krein-like theorems regarding signature and bifurcation areproven. In addition, a canonical Hamiltonian example, composed of a negativeenergy oscillator coupled to a heat bath, is discussed and our development iscompared to previous work in this context. A careful counting of eigenvalues , withand without symmetry, is undertaken, leading to the definition of a degeneratecontinuum steady-state (CSS) bifurcation . It is described how the CHH and CSSbifurcations can be viewed as linear normal forms associated with the nonlinearsingle-wave model described in [BAL13], which is a companion to the present workand that of Chapter 12.

Chapter written by George 1. HAGSTROM and Philip 1. MORRISON.

284 Nonlinear Physical Systems

13.1. Introduction

Bifurcations of unstable modes from the continuous spectrum underlie patternformation in a wide variety of physical systems that can be described by Hamiltonian2+ I field theories. These patterns take the form of vortices in phase space , and arereferred to as "BGK modes" (Bernstein, Greene, Kruskal) in plasma physics[BER 58] and "Kelvin cat's-eye" vortices in two-dimensional, inviscid ,incompressible fluid mechanics, and possess analogs in condensed matter physics,geophysical fluid dynamics and astrophysics. The equations that describe thesephysical systems all share crucial features : a formulation as a noncanonicalHamiltonian system [MOR 82, MOR 98] and that stable equilibria possesscontinuous spectra. Before nonlinear patterns form in these systems, unstable modesbifurcate from their continuous spectra, a linear bifurcation we call the CHHbifurcation that is an analog of the usual HH bifurcation of finite-dimensionalsystems. In this chapter, we describe some mathematical aspects of the CHHcontinuing on from the material presented in Chapter 12.

Perturbation of point spectra in canonical, finite-degree-of-freedom Hamiltoniansystems is described by Krein's theorem [KRE 50, KRE 80, MOS 58] , which statesthat a necessary condition for an HH bifurcation is to have a collision betweeneigenvalues of opposite signature. A different situation arises in theinfinite-dimensional case if the linear Hamiltonian system has a continuous spectrum.A representative example of such a Hamiltonian system is the Vlasov-Poissonequation [MOR 80], which when linearized about stable homogeneous equilibriagives rise to a linear Hamiltonian system with pure continuous spectrum that can bebrought into action-angle normal form [MOR 92, MOR 94a, MOR 94b, MOR 00]. Adefinition of signature was given in these works for the continuous spectrum. Theprimary example here will be the Vlasov-Poisson equation, but the same structure ispossessed by a large class of equations [MOR 03], examples being Euler's equationfor the two-dimensional fluid, where signature for shear flow continuous spectra wasdefined [BAL 99, BAL 01] , and likewise for a model for electron temperaturegradient turbulence [TAS 11]. Modulo technicalities, the behavior discussed here, isexpected to cover a large class of systems.

In section 13.2, we present the mathematical structure that we use to describeCHH bifurcations; in particular, we define structural stability and discuss thedefinition of signature for the continuous spectrum. One of the crucial parts of thisframework is the choice of the norm on the perturbations to the time evolutionoperator, a step that requires selection of a Banach space to be the phase space forsolutions of the linearized system. In section 13.3, we apply this framework to theVlasov-Poisson equation, presenting without proof the results that appeared in[HAG 11b]. We show that the plasma two-stream instability is a CHH bifurcationthat can be viewed as a zero-frequency mode interacting with a negative energycontinuous spectrum to bifurcate to instability so that the continuous spectrum

provides the "other" mode in theBanach space, the sup of the Eequilibria of the Vlasov-Poissoiexamples of such Banach spaceperturbations to those that are d:the possibility of altering the sigequilibria with positive signature I

Section 13.4 contains a descnoncanonical systems; in particucanonical Hamiltonian systems isto instability in such a canonical :idea that a certain mean fieh[TEN 94, DEL 98, BAL 13], is a j

describes the eventual nonlinear sWe note that this model is deriveHamiltonian 2+ 1 mean field theotcomparison with the results asection 13.6, we summarize and c

13.2. Mathematical aspects of tl

In section 12.3.2.1 of Chapteibifurcation, the plasma two-strcondition for structural instabilitypresent a framework for bifurcacontinuous spectra. The key not:continuous spectrum, which iHamiltonian systems that undergdiscussing structural stability.

13.2.1. Structural stability

Now, we consider structural stwith continuous spectrum. The dyfunction space fjg . We are given a Iis (typically) an unbounded quad]bracket {. , .}, which will be bilinand which in this chapter will alwaHamilton's equations are (see Cha

St = {s ,H} = 'IS .

tinuous spectrum underlie patternIt can be described by Hamiltonianf vortices in phase space , and arene, Kruskal) in plasma physicsin two-dimensional, inviscid,

ogs in condensed matter physics,he equations that describe these

formulation as a noncanonicalthat stable equilibria possess

l in these systems, unstable modesir bifurcation we call the eHRbifurcation of finite-dimensionalithematical aspects of the CRRer 12.

:e-degree-of-freedom Hamiltonian, KRE 80, MOS 58], which statesn is to have a collision between~rent situation arises in thesystem has a continuous spectrum.n system is the Vlasov-PoissonIt stable homogeneous equilibria~ continuous spectrum that can be({OR 94a, MOR 94b, MOR 00]. Afor the continuous spectrum. Theequation , but the same structure is, examples being Euler's equation;hear flow continuous spectra wasmodel for electron temperature

~s, the behavior discussed here , is

Continuum Hamiltonian Hopf Bifurcation II 285

provides the "other" mode in the CHH bifurcation. We show that if in the chosenBanach space, the sup of the Hilbert transform, is an unbounded operator, thenequilibria of the Vlasov-Poisson equation are always structurally unstable. Twoexamples of such Banach spaces are W 1,1(JR) and CO (JR) n L1(JR). If we restrictperturbations to those that are dynamically accessible [MOR 98], which precludesthe possibility of altering the signature of the continuous spectrum, we prove thatequilibria with positive signature only are structurally stable.

Section 13.4 contains a description of the differences between canonical andnoncanonical systems; in particular, comparison to the work of KreIn [DAL 70] oncanonical Hamiltonian systems is made and a simple demonstration of a bifurcationto instability in such a canonical system is described. In section 13.5, we present theidea that a certain mean field Hamiltonian system, the single-wave model[TEN 94, DEL 98, BAL 13], is a nonlinear normal form for the CHH bifurcation thatdescribes the eventual nonlinear saturation of the resulting instability near criticality.We note that this model is derived by means of matched asymptotic expansions of aHamiltonian 2+1 mean field theory near a marginally-stable equilibrium, and also bycomparison with the results of numerical simulations [BAL 13]. Finally, insection 13.6, we summarize and conclude.

13.2. Mathematical aspects of the continuum Hamiltonian Hopf bifurcation

In section 12.3.2.1 of Chapter 12, we presented a specific example of the CHHbifurcation, the plasma two-stream instability. This theory gives a necessarycondition for structural instability: collision of eigenvalues of opposite signature. Wepresent a framework for bifurcations in noncanonical Hamiltonian systems withcontinuous spectra. The key notion will be the generalization of signature to thecontinuous spectrum, which is prevalent in the linear infinite-dimensionalHamiltonian systems that undergo the CHH bifurcation. We first set the stage bydiscussing structural stability.

13.2.1. Structural stability

Now, we consider structural stability of linear noncanonical Hamiltonian systemswith continuous spectrum. The dynamical variable Sis assumed to be a member of afunction space ~. We are given a Hamiltonian functional H (HL of Chapter 12), whichis (typically) an unbounded quadratic functional on ~, and a noncanonical Poissonbracket {. , .}, which will be bilinear, antisymmetric, and satisfy the Jacobi identityand which in this chapter will always be of Lie-Poisson form, see [MOR 98, MAR 99].Hamilton's equations are (see Chapter 12)

structure that we use to describeictural stability and discuss the1. One of the crucial parts of this.turbations to the time evolution1 space to be the phase space for, we apply this framework to theoof the results that appeared ininstability is a CHH bifurcation

teracting with a negative energy;0 that the continuous spectrum Sf = {S,H} = 'IS· [13.1]


Here, '1' is the time evolution operator, which by assumption is a linear operatorfrom /18 to itself. This equation can also be written as

[13.2]

where J is the cosymplectic operator of the bracket {' ,'}, and s.2l is a linear operatorderived from H using the bracket. Care must be taken when using this formulation asthe operator s.2l is often not defined as an operator on /18, and only the product Js.2l = '1'takes values in /18. The process of canonization, which reformulates equation [13.2]in terms of a canonical cosymplectic operator Je, which is bounded and invertible,eliminates this difficulty.

The operator '1' (and hence the linear Hamiltonian system) is spectrally stable ifthe spectrum of '1' is contained in the imaginary axis, <J"('T) E iJR. This is equivalent tothe condition that the spectrum is in the closed lower half plane, i.e. Re( <J"('T)) :::; 0,because the spectrum satisfies the property A E <J"('T) implying XE <J"('T), a propertythat comes from the Hamiltonian structure. Solutions of spectrally stable systems growat most sub-exponentially.

We consider now a family of Hamiltonian systems that depend continuously onsome parameter and look for changes in the stability of the Hamiltonian system asthe parameter varies. Such a family can be generated in many ways. One commonscenario is for the linear Hamiltonian system to come from the linearization of somenonlinear Hamiltonian system about an equilibrium solution. In that case, the bracketand Hamiltonian functional come from linearizations of the original Hamiltoniansystem, and both will depend on the equilibrium fa (see Chapter 12). Both thebracket J and Hamiltonian H are subject to change, however, and this malleability ofthe bracket gives the bifurcation theory of noncanonical Hamiltonian systems adifferent character than that of canonical Hamiltonian systems.

Bifurcations to instability occur when a spectrally stable system becomesspectrally unstable. The following definitions depend on our definition of the size ofa perturbation, and we will have to choose some set of perturbations and a measure ofthis size in order to proceed. Assuming that we have made this choice, we say thatthe spectrally stable time evolution operator '1' is structurally stable if there existssome £ such that for all perturbations 8'1' satisfying 11 8'1' 11 < e, the operator '1' + 8'Tis spectrally stable, where II . II is our chosen measure for perturbations of 'T.Otherwise, we say that '1' is structurally unstable.

The theory will depend on the choice of allowable perturbations and norm. Letthe family of Hamiltonian systems be parametrized by a parameter A so that the timeevolution operator for each system is '1'A' The continuity properties of the family 'TA

will typically come from an indueto the equations live. For instancoperator. Other choices are relati[GRI 90] in the context of canonicperturbations based on the gap, gi

Some of the most physicallinearizing a Hamiltonian syste:equilibrium states. If this is the conly consider perturbations thatinstance restricting ourselves to piOur most important example wiVlasov-Poisson system, which isperturb to equilibria that are dyn:which restricts us to perturbations

13.2.2. Normalforms and signau

The KreIn signature is essentcollision between positive and neg;existence of a HH bifurcation. It iof modes in the finite-dimensionfunction. In the infinite-dimensioncontinuous spectrum. The continugeneralized eigenfunctions, are di:another approach, based on the 1

signature to the continuous spectru

The linear theory of finite-dimenormal forms , and the proof of Kforms. Though the situation can bemany important cases, it is possiblenormal forms arise when the timediagonalizable, in which case the Hvariables (e(u),J (u)) and a canoni:

H = tdU<J"(u) ro(u)J (u) = ~

Wherer c JR, and in the second equthe canonically conjugate oscillatorrand o («) E {±1} defines the sigto iro(u).


by assumption is a linear operatoras

[13.2]

will typically come from an induced norm from the Banach space in which solutionsto the equations live. For instance, one choice would be that 'I - 'IA, is a boundedoperator. Other choices are relatively compact/bounded perturbations (considered in[GRI 90] in the context of canonical systems) or the more general class of unboundedperturbations based on the gap, given by [KAT 66].

13.2.2. Normalforms and signature

The linear theory of finite-dimensional Hamiltonian systems is organized aroundnormal forms , and the proof of Krein's theorem is based on the theory of normalforms. Though the situation can be more complex in the infinite-dimensional case , formany important cases, it is possible to find the appropriate normal form. The simplestnormal forms arise when the time evolution operator of the Hamiltonian system isdiagonalizable, in which case the Hamiltonian can be written in terms of action-anglevariables (e(u),J (u)) and a canonical Poisson bracket, for instance

Some of the most physically interesting families of systems come fromlinearizing a Hamiltonian system about a member of a continuous family ofequilibrium states. If this is the context of our physical problem, it makes sense toonly consider perturbations that leave this Hamiltonian structure unchanged, forinstance restricting ourselves to perturbations that change the equilibrium state only.Our most important example will be the two-stream instability described by theVlasov-Poisson system, which is of this type. A further restriction is to choose toperturb to equilibria that are dynamically accessible from the original equilibrium,which restricts us to perturbations that can be produced using Hamiltonian forces .

The Krein signature is essential in the description of HH bifurcations as thecollision between positive and negative energy modes is a necessary condition for theexistence of a HH bifurcation. It is straightforward to compute the energy signatureof modes in the finite-dimensional case as we can simply use the Hamiltonianfunction. In the infinite-dimensional case, this is complicated by the presence of thecontinuous spectrum. The continuous analogs of discrete eigenmodes, the so-calledgeneralized eigenfunctions, are distributions whose Hamiltonian is not defined, andanother approach, based on the theory of normal forms, is required to attach asignature to the continuous spectrum.

[13.3]H = rdu <J"(u) m(u )J(u) = ~ rdu o («) m(u) (Q(u) 2+P(u )2) ,Jr 2 Jr

iectrally stable system becomesnd on our definition of the size of.of perturbations and a measure ofive made this choice, we say thatstructurally stable if there existsg 118'I11< £, the operator 'I+ 8'Imeasure for perturbations of 'I.

tems that depend continuously onlity of the Hamiltonian system asated in many ways. One commonme from the linearization of some1 solution. In that case, the bracket:ions of the original Hamiltoniann fa (see Chapter 12). Both the, however, and this malleability of.anonical Hamiltonian systems aan systems.

iian system) is spectrally stable if.s, <J"('I) E iJR. This is equivalent tower half plane, i.e. Re( <J"('I)) ~ 0,'I) implying X E <J"('I), a propertylS of spectrally stable systems grow

et {. , .}, and Qt is a linear operator(en when using this formulation asn ~, and only the product JQt = 'Izhich reformulates equation [13.2]which is bounded and invertible,

'able perturbations and norm. Letby a parameter A so that the time

inuity properties of the family 'IA,

where I" c JR, and in the second equality, we introduce the alternative form in terms ofthe canonically conjugate oscillator variables Q(u) and P(u). Here, to E JR>ofor all u E

rand <J"(u) E {±I} defines the signature of the continuous spectrum correspondingto im(u).


The ability to define the signature for a given Hamiltonian system is directlyrelated to the ability to bring the system into a normal form, i.e. to canonize anddiagonalize it. Diagonalization is equivalent to finding a transformation that convertsthe time-evolution operator of the system into a multiplication operator, viz., tofinding a linear operator .£ such that .£'I.£-l is a multiplication operator. The systemsdescribed in this chapter all tend to have non-normal time-evolution operators; so, itmay be surprising that it is ever possible to define such a transformation since thespectral theorem does not apply, but it turns out that for many important cases, thetime evolution operators are diagonalizable. Operators with this property are calledspectral operators [DUN 88]. By definition, a spectral operator possesses a family ofspectral projection operators <E( (5) defined on Borel subsets <5 c C that commutewith the time evolution operator 'I and reduce its spectrum, i.e. cr('I<E((5)) c <5.

The signature of the subset <5 is then defined by the sign of the Hamiltonianoperator restricted to members of <E( (5)88, which can be positive , negative orindefinite. For a given point u E C, this is defined by taking limits of sets <5 thatcontain u. If a diagonalization is known, the application of this definition can bestraightforward . Consider equation [13.3] with crm(u) = u, u E IR>o. Then, <E(<5)88 isequal to the functions with support on IR n (-i<5) and the energy is clearly positive.An equivalent definition involves the sign of the operator .J on the spectrum but thedefinition involving signature is more intuitive physically.

13.3. Application to Vlasov-Poisson

Now, consider the Vlasov-Poisson system of section 12.3.2.1 of Chapter 12, as anexample of the Hamiltonian 2+1 field theories that exhibits the CHH bifurcation . Here,we are interested in the properties of the equations linearized around a homogeneousequilibrium fa. These equations , repeated for completeness, are

by II <5f~ II in the norms for W 1,1 aneis a bounded operator from those s

If we were to consider L2 (IR) , 1

would have to use some other mealKato [KAT 66]).

As mentioned earlier, the lim:spectrum. Morrison [MaR 00] cclinearized Vlasov-Poisson systenmultiplication operator and deten(see [MaR 03] for a generalizatietheories). This is based on the Gdiscrete modes is defined as follow

where 6[f] operating on a function

and the Hilbert Transform is define

[13.4] £ [g]= ~1 dp g(p) ,nJR p-u

Our goal is to understand how the spectrum of 'Ik changes under changes in fa·To this end, we consider perturbations of the equilibrium function fa + <5 fa. The timeevolution operator of the perturbed system is 'Ik+ <5'Ik, where

where the expression :hR stands fclinearized Vlasov-Poisson equatior

[13.5]dgk- -v ikugv =0dt '

We use the operator norm induced by the norm on 88 to measure the size of <5'Ik'which requires restriction to function spaces in which <5'Ik is bounded, for instance,the Sobolev spaces WI ,1 (IR) and en (IR) nL 1 (IR). The quantity II <5'IkII can be bounded

Which gives a representation of the

Using this G-transform and thecanonize and diagonalize the Iii


by 118f~ IIin the norms for WI,1 and en nL1 because the integral operator fIRdPSk (p,t)is a bounded operator from those spaces to IR

If we were to consider L2(IR), then 8'Ik would be an unbounded operator and wewould have to use some other means of determining its size (see Grillakis [GRI 90] orKato [KAT 66]).

[13.6]

[13.7]and

As mentioned earlier, the linearized Vlasov-Poisson system has a continuousspectrum. Morrison [MOR 00] constructed a transformation that diagonalizes thelinearized Vlasov-Poisson system, converting the time evolution operator to amultiplication operator and determining the signature of the continuous spectrum(see [MOR 03] for a generalization of this method to other 2+ 1 Hamiltonian fieldtheories). This is based on the G-transform, which for stable equilibria fa with nodiscrete modes is defined as follows:

i by the sign of the Hamiltonianich can be positive, negative ored by taking limits of sets 8 thatolication of this definition can be:(u) = u,u E IR>o. Then, ~(8)36' isand the energy is clearly positive.iperator J on the spectrum but thesically.

~n Hamiltonian system is directlynormal form, i.e. to canonize andding a transformation that converts1 multiplication operator, viz. , to.ultiplication operator. The systemsnal time-evolution operators; so, itre such a transformation since the:hat for many important cases, theators with this property are called.tral operator possesses a family ofrrel subsets 8 c C that commutepectrum, i.e. (J('I~(8)) c 8.

where O[f] operating on a function f in its domain is the inverse of G,

ction 12.3.2.1 of Chapter 12, as an.xhibits the CHH bifurcation. Here ,linearized around a homogeneous

ileteness, are and the Hilbert Transform is defined as

[13.8]

[13.4] £ [g] = ~1 dp g(p ) ,n:JIR p-u

[13.9]

: 'Ik changes under changes in fa.brium function fa + 8fo. The timeS'Ik, where

where the expression fJR. stands for the Cauchy principal value. If !k satisfies thelinearized Vlasov-Poisson equation, then gk = O[fk] satisfies

[13.5][13.10]

on 36' to measure the size of 8'Ik,rich 8'Ik is bounded, for instance,re quantity 118'Ik ll can be bounded

which gives a representation of the solution upon back-transforming.

Using this G-transform and the theory of generating functions, it is possible tocanonize and diagonalize the linearized Vlasov-Poisson system. Canonization


proceeds by transforming the Poisson bracket of equation [12.35] of Chapter 12according to

where o'(u) = sgn( UtI) is the sigrOJ = Ikul.

The canonically conjugate pair (22k(p), &k(p) are real due to the reality of thedistribution function f = fa + S, and the Poisson bracket in terms of them is

r (8F 8G 8G 8F){F,G} = JIR dp 822 8& - 822 8& '

wherein, and often henceforth, we suppress the kEN dependence.

[13.11]

[13.12]

The Hamiltonian of [13.17],oscillators, is the normal fonn 1transformation can be defined onlyalways be made true by Galilean ~

the sign of uf~ changes. To illustn

of a Maxwellian distribution, fa =fa = e-(P-Pl)2 + e-(p+pd2 (where

distribution has one maximum, antother hand , the bi-Maxwellian haschanges.

Diagonalization is achieved using a mixed-variable generating function involvingthe G-transfonn, a transformation that was inspired by Van Kampen's formalexpression for continuum eigenmodes [VAN 55] 0.8

0.6

'Ik"Y(U ,p ) = iku"Y(u,p). [13.13]0.4

where 0.2

Positive Signature01---------.---

-0.2[13.14]

1"Y(u,p) := tI(p)PV- + tR(p)8(u - p) ,

u-v

which clearly bares the mark of the G-transform. Diagonalization proceeds from thefollowing mixed-variable generating functional [MOR 00]

-0.4

-0.6

-0.8

-4-6-1 '-----'-------"----'-----10 -8

[13.15]

which leads to a transformation to the new variables (Q,P)Figure 13.1. Signature [c

Under direct substitution into the Hamiltonian and making use of identitiesderived in [MOR 92, MOR 00] (see also [MOR 07]) in terms of the new variables,the Hamiltonian becomes

Q = 8.%2 = 0[22]8P

[13.16]

[13.17]

The Penrose criterion, which w:the role that signature plays in traPoisson system. This criterion is tlline under the map tR + itI is equafamily of equilibria that depend ccSome values of the parameter andchange , the Penrose plot must incr


rf equation [12.35] of Chapter 12 where a(u) = sgn(ut]) is the signature of the continuous spectrum with frequency

to> Ikul·

[13.11]

I) are real due to the reality of therracket in terms of them is

[13.12]

:: N dependence.

The Hamiltonian of [13.17], that for a continuum of uncoupled harmonicoscillators , is the normal form for the linearized Vlasov-Poisson system. Thistransformation can be defined only in reference frames where f~(O) = 0, which canalways be made true by Galilean shift. Therefore, the signature changes only whenthe sign of uf~ changes. To illustrate this signature, consider two special cases, that

of a Maxwellian distribution, fo = e-p2, and that of a bi-Maxwellian distribution,

fo = e -(p-p\)2 + e-(p+pd2

(where normalization is not important). The Maxwelliandistribution has one maximum, and therefore, it has only positive signature. On theother hand, the bi-Maxwellian has three extrema (see Figure 13.1) and two signaturechanges .

Positive Signature

Negative Signature

Positive Signature01----------=------''

0.2

-0.4

-0.2

-0.6

0.8

0.6

0.4[13.13]

[13.14]

Diagonalization proceeds from thelOR 00]

iable generating function involvingspired by Van Kampen's formal

-0.8

108642o-2-4-6-1 L..-.-_-'-_-----'__--'--_------l.__--'--_----'-__--'--_--'-__-'-_---J

-10 -8

[13.15]

es (Q,P)Figure 13.1. Signature for a bi-Maxwellian distribution function

"[Pl · [13.16]

ian and making use of identities)7]) in terms of the new variables ,

[13.17]

The Penrose criterion, which was introduced in Chapter 12, clearly demonstratesthe role that signature plays in transitions to instability for the linearized VlasovPoisson system. This criterion is that the winding number of the image of the realline under the map tR + it] is equal to 0 when fo is stable. Suppose that we have afamily of equilibria that depend continuously on some parameter, and is stable forsome values of the parameter and unstable for others . In order for the stability tochange , the Penrose plot must increase its winding number from 0 to 1. For this to


happen, the Penrose plot must cross the origin at the bifurcation point. We call thesecrossing points critical states.

A simple technique to compute the winding number is to draw a ray from theorigin to infinity and to count the number of intersections with the contour, accountingfor orientation by adding 1 for a positive orientation and subtracting 1 for a negativeorientation. We count the number of zeros of C[ for which CR < 0 and add them with apositive sign if f6' is positive, a crossing of the Penrose plot from the upper half planeto the lower half plane; a negative sign if f~ is negative, a crossing from the lowerhalf plane to the upper half plane; and zero if u is not a crossing of the real axis, atangency.

13.3.1. Structural stability in the space en(IR) nL l (R)

The first critical state occurs wAt such a point, the addition of aplot to intersect the real axis trarcause instability. If the system istransverse intersections, then the wand the system would be unstable.a bifurcation at k =I- O.

Another critical state occurstransversely intersects the real axi:will be a crossing with a negativeFigure 13.3 is a critical Penrose plwith the maximum stable separatic

for some constant e independent of 8 f6. Therefore, for fixed k, any stable fa that doesnot contain a discrete mode is structurally stable as the Penrose plot will be a fixeddistance from the origin.

Furthermore, the other part of the Penrose plot, 1 - nk- 2£ [f6] , is bounded in asimilar way because the sup norm of the Hilbert Transform of 8 f6 is bounded by theen norm of 8f6, viz

We begin by choosing the phase space of the linearized Vlasov-Poisson system tobe en(R) nL1(iR). In this phase space, the induced norm on 8'I is proportional to thesup of 8 f6. This choice puts a restriction on the ability of perturbations to affect thesignature of f6; at a point u, a perturbation must have norm at least f6(u) to induce asignature change at u, viz Consider the k =I- 0 bifurcatior

frequency mode. Then, we claim tJCRI Ju<O. This is demonstrated :dispersion function c. Suppose thethen a small perturbation would geincrease it. This would imply the ethe analyticity of the plasma disp:the inflection point mode is alwacontinuous spectrum. The fact thatsignature may seem counterintuiti:can be added in the neighborhoodperturbation. When we restrict 01

which we do in a later section, this

To understand and interpret theof the embedded modes at the critienergy signature of an embeddesgn(uJcrIJu) [MOR 92, SHA 94

In the k = 0 bifurcation, the si~

spectrum surrounding it are alwafrequency, then the embedded modis negative (the embedded mode iWhich can be seen again using ana"section (see [HAG llb])), or therspectrum. There is no referencespectrum and embedded mode are

[13.19]

[13.18]

Csup 1£[8f6] I ::; II 8'Ik II ,

When fa has an embedded mode , it is possible to have transitions to instability.We identify two critical states for the Penrose plots that correspond to the transition toinstability. In each of these states, there is an embedded mode inside the continuousspectrum. In the first state , the embedded mode is a so-called inflection point mode[SHA 94], which means there is some rocIk = Uc such that CR(uc ) = C[(uc ) = 0 andf6' (uc ) = O. We refer to this state as the bifurcation at k =I- 0 because changing thevalue of k would not cause a bifurcation. In the other state, f6' =I- 0, which we call thebifurcation at k = O. This is named so that in a system with infinite spatial extent, theunstable mode first appears at k = O.

he bifurcation point. We call these

rumber is to draw a ray from thections with the contour, accounting.m and subtracting 1 for a negativewhich CR <°and add them with a

rose plot from the upper half planeegative, a crossing from the lowers not a crossing of the real axis, a

,1(R)

iearized Vlasov-Poisson system tonorm on <5'I is proportional to the

oility of perturbations to affect therve norm at least 16 (u) to induce a

[13.18]

, 1 - ntc?£[/6], is bounded in aransform of <516 is bounded by the

[13.19]

, for fixed k, any stable 10 that does1S the Penrose plot will be a fixed

~ to have transitions to instability.that correspond to the transition toedded mode inside the continuousa so-called inflection point mode

such that cR(Uc ) = c/(uc ) = °andon at k i- °because changing theer state, I~ i- 0, which we call theem with infinite spatial extent, the


The first critical state occurs when 16(u) = 0, 16'(u) = °and 1- nk-2£ [/6] = 0.At such a point, the addition of a generic function <510 to 10 will cause the Penroseplot to intersect the real axis transversely, and such a perturbation can be used tocause instability. If the system is perturbed so that the tangency becomes a pair oftransverse intersections, then the winding number of the Penrose plot would jump to 1and the system would be unstable. Figure 13.3.1 illustrates a critical Penrose plot fora bifurcation at k i- 0.

Another critical state occurs when 1 - nk-2£[/6] = °at a point where 16transversely intersects the real axis. If the Hilbert transform of 16 is perturbed, therewill be a crossing with a negative £[/6], and the winding number will be positive.Figure 13.3 is a critical Penrose plot corresponding to the bi-Maxwellian distributionwith the maximum stable separation.

To understand and interpret these bifurcations, we must understand the signatureof the embedded modes at the critical state and also of the continuous spectrum. Theenergy signature of an embedded mode with frequency U = (j)Ik is given bysgn(udcrldu) [MOR 92, SHA 94].

Consider the k i- °bifurcation. Assume that the embedded mode is not a zerofrequency mode. Then, we claim that if Uc > 0, then dCRIdU > °and if Uc < 0, thendCRI dU<O. This is demonstrated in [HAG lIb] using the analyticity of the plasmadispersion function c. Suppose the opposite were true, that Uc > °and dCRIdu < 0,then a small perturbation would generically decrease the winding number rather thanincrease it. This would imply the existence of poles in the upper half plane, violatingthe analyticity of the plasma dispersion function. This implies that the signature ofthe inflection point mode is always the same as the signature of the surroundingcontinuous spectrum. The fact that this bifurcation occurs when there is only positivesignature may seem counterintuitive, but it is due to the fact that negative signaturecan be added in the neighborhood of the inflection point mode with an infinitesimalperturbation. When we restrict ourselves to dynamically accessible perturbations,which we do in a later section, this bifurcation will disappear.

In the k = °bifurcation, the signature of the embedded mode and the continuousspectrum surrounding it are always indefinite, either the embedded mode is at °frequency, then the embedded mode has zero energy and the signature surrounding itis negative (the embedded mode is always in a valley of the distribution function,which can be seen again using analyticity and the perturbation introduced in the nextsection (see [HAG lIb])), or there is a change in the signature of the continuousspectrum. There is no reference frame in which the signature of the continuousspectrum and embedded mode are definite.


0.8

0.6

0.4

0.2

~I

-0.2

-0.4

-0.6

- 0.8

- 1- 1.5 - 1 - 0.5

Figure 13.3. Critical Penrose .

1hp/£h sgn(p)

X = h +d/2+£/2 - p/2-h - d/2 - £/2 - pio

Suppose we perturb 10 by aoperator T, is the operator mappinbounded operator and thus 11 81611 ~

perturbations that can be made inunity. For example , consider the fur

We explicitly demonstrate this ~

and, by extension , the Banach spastable distribution function is strucaffairs.

to an infinitesimal perturbation. Ioperator, i.e. there exists an infiniteof 16. Such a perturbation can n£ [/6+ 816] > 0 as well - therelzeros of the Penrose plot and causii

0.8

0.6

0.4

0.2

' 0....., 0I

- 0.2

-0.4

-0.6

-0.8

-1-0.6 -0.4 - 0.2 0 0.2 0.4 0.6 0.8 1.2

H [!o]

(a)

0.4

0.3

0.2

0.1

~ 0I

- 0 .1

-0.2

-0.3

-0.4

-0.5- 0.6 - 0.5 - 0.4 - 0.3 -0.2 -0.1 0.1 0.2 0.3

H [fo]

(b)

Figure 13.2. a) Critical Penrose plot for a k =1= 0 bifurcation. b) Close-up ofpanel a

13.3.2. Structural stability in Wi ,I

We will prove that if the perturbation function is some homogeneous 810 and thespace is WI ,1 (IR), then every equilibrium distribution function is structurally unstable

Figures 13.4(a) and (b) show trespectively. In the space WI ,I , (IR)We choose h = d and £ = O(e- I / h ) ,


to an infinitesimal perturbation. Under this choice, sup 1£[f6] I is an unboundedoperator, i.e. there exists an infinitesimal 8 fo such that £ [8f6]is order one at a zeroof f6. Such a perturbation can tum any point where f6 = 0 into a point whereYt'[f6 + 8 f6] > 0 as well - thereby changing the winding number by moving thezeros of the Penrose plot and causing a bifurcation to instability.

0.6 0.8 1.2

0.8

0.6

0.4

0.2

-..::; 0I

-0.2

-0.4

-0.6

-0.8

- 1-1. 5 - 1 -0.5 2.5

HUn]

Figure 13.3. Critical Penrose plotfor a bi-Maxwellian distribution function

We explicitly demonstrate this structural instability for the Banach space Wl ,l (IR)and, by extension, the Banach space L 1(IR) nCO (IR), and this will imply that everystable distribution function is structurally unstable, a physically unappealing state ofaffairs.

Suppose we perturb fo by a function 8 fo. The resulting perturbation to theoperator T, is the operator mapping Sk to 8 f6 Jdp Sk. In the space Wl ,l (IR) this is abounded operator and thus 118f611 ~ 118'Ikll. Yet, it is possible to introduce a class ofperturbations that can be made infinitesimal, but have Hilbert transform of orderunity. For example, consider the function X(p,h,d, £) defined by

Ipl < ee < Ipl < d-i-e2h+d+£ > p > d-v e2h+d+£ > -p > d-i-eIpl > 2h+d+£1

hp ieh sgn(p)

X = h+d/2+£/2- p/2-h - d/2 - £/2 - p/2o

0.3

urcation. b) Close-up ofpanel a

s some homogeneous 8 fo and the,n function is structurally unstable

Figures 13.4(a) and (b) show the graph of X and its Hilbert transform, £ [X ],respectively. In the space Wl,l , (IR) the function X has norm 2h2 + 2hd + he +4h. Ifwe choose h = d and e = O(e- l / h ) , then the terms that do not involve £ are all smaller


than (6h+ s ) log(6h+e). With these choices, X satisfies

X(O) 2 - (h+e- 1/h

) log(lh+e-1/h I)

+(3h +e- 1/h

) log(13h +e- 1/h I)

2+0(hlogh). [13.20]

that are dynamically accessible fnthe mean-field Hamiltonian fieldwritten as a composition an initialZ (q,p) describes the single particlfunctions fl and h are dynamicalsymplectic map Z such that fl =and vice versa.

0.15...------.-----r--- ,----

If we choose d = hand e = e-(I /h), then for any 8, r > 0, we can choose anh such that Ilxlll,1 < 8 and fdpX/p > 1- O(h) and Ifdpx/(u- p)1 < Ir/ul forlui> 12h+d+£I·

The perturbation X arbitrarily moves the crossings of the real axis of the Penroseplot of fo. If we use this perturbation to move crossings from the positive imaginaryaxis to the negative real axis, we can increase the winding number of the Penrose plot,thus causing instability. Therefore the existence of this X implies that any equilibriumis structurally unstable in both the spaces W 1,1(JR) and L1(JR) n CO (JR).

0.1

0.05

-0.05

-0.1

THEOREM 13.1.- A stable equilibrium distribution fo is structurally unstable underperturbations of the equilibrium in the Banach spaces WI ,1(JR) and L 1(JR) nCO (JR).

- 1 -0.8 -0.6 -0.4

Thus, we emphasize that we can always construct a perturbation to fo that makesour linearized Vlasov-Poisson system unstable. For the special case of theMaxwellian distribution, Figure 13.5(a) shows the perturbed derivative of thedistribution function and Figure 13.5(b) shows the Penrose plot of the unstableperturbed system. Observe the two crossings created by the perturbation on thepositive axis as well as the negative crossing arising from the unboundedness of theperturbation.

Theorem 13.1 expresses the fact that in the norm W 1,1, signature changes giverise to unstable modes under infinitesimal perturbations combined with the fact that asignature change can be induced in the neighborhood of any maximum of fo. In thenext section, we will demonstrate the role of signature more explicitly by restrictingto dynamically accessible perturbations .

13.3.3. Dynamical accessibility and structural stability

As we have stated, the signature of the continuous spectrum in theVlasov-Poisson system is sgn(u£](u)). In W 1,I(JR ), sUPl8f61 is bounded by 1/8f61/,which means that most points of the continuous spectrum cannot change signatureunder infinitesimal perturbations, the exception being near points where f6 vanishes.All signature changes can be prevented by restricting ourselves to perturbations of fo

2

1.5

:E:t:

0.5

-J ------! ! !

- 1 -0.8 -0.6 -0.4

Figure 13.4. a) The perturbati

In this chapter, we only study Iimpossible to construct a dynamic;domain that preserves spatial hom.


tisfies

[13.20]

that are dynamically accessible from fa as we shall explain. The solution of any ofthe mean-field Hamiltonian field theories that have been described here can bewritten as a composition an initial condition Jwith a symplectic map Z(q,p), whereZ(q,p) describes the single particle characteristics (see [MOR 90]) . We say that twofunctions hand h are dynamically accessible from each other if there exists somesymplectic map Z such that fl = h 0 Z, i.e. h is a symplectic rearrangement of I:and vice versa.

any 8,r > 0, we can choose anand If dp X/ (u - p) I < Ir/uI for

ngs of the real axis of the Penrose:sings from the positive imaginaryinding number of the Penrose plot,.his X implies that any equilibriumand L 1(IR) n CO (IR) .

1 fa is structurally unstable underes W I ,1(IR) and L1(IR) n CO (IR) .

0.15

0.1

0.05

><

- 0.05

-0.1

- 1 -0.8 - 0.6 -0.4 - 0.2

(a)

ov

0.2 0.4I

0.6I

0.8

T2 ,--- ----,-----r----.-------..,--- ---,,--- ----,-- - --r-- - ---,----,------,

1.5 f-

ret a perturbation to fa that makes., For the special case of thethe perturbed derivative of the:he Penrose plot of the unstable~ated by the perturbation on the19 from the unboundedness of the

irm W 1,1, signature changes giveions combined with the fact that a.od of any maximum of fa. In theture more explicitly by restricting

0.5

I

- 0 .8!

- 0 .6,===

- 0.4 - 0.2 0.2 0.4I

0.6I

0.8

(b)

~ility

~ continuous spectrum in the), sup 1 8f~1 is bounded by 11 8f~l l ,

pectrum cannot change signatureng near points where f~ vanishes .19 ourselves to perturbations of fa

Figure 13.4. a) The perturbation xfor c = e-1a ,h = d = 0.1. b) The Hilberttransform ofX

In this chapter, we only study perturbations of fa that preserve homogeneity. It isimpossible to construct a dynamically accessible perturbation of fa in a finite spatialdomain that preserves spatial homogeneity. To see this , we write a rearrangement as


Figure 13.5. a) Perturbed Maxwellian distribution, f6 +x. b) The Hilberttransform ofpanel a

(q,p) f-t (q,p), where p is a function of p alone. Because [q,p] = I and p(p) is nota function of q, we have p'oq/oq = 1 or q = q/ p'. If the spatial domain is finite , thismap is not a diffeomorphism unless p' = 1. For infinite spatial domains, this is not aproblem and these rearrangements exist.

These ideas lead to the followinperturbations in the WI ,1 norm:

THEOREM 13.2.- Let fa be a stableequation on an infinite spatial ddynamically accessible perturbatioif~ (p) = O. If there are multiple solumodes come from the zeros of f~ th:

accessible perturbations cannot chnumber of critical points of fa. Indeand p is a homogeneous, i.e . a fun'are the points p-l (Pc), where Pc iscritical points wilbalways be the salthe map pmust be monotonically in

If Pc is a non-degenerate criti:previous obstruction to the applperturbation does not apply. In [HAp such that fa(p)dfo(p) /dp = fMp) + X(p - Pc) . Suthe parameters defining X, thf~ (p) +X(p - Pc) has the same crittheory to find a p so that fa(p) = .has compact support and is smaller 1

One implication is that the peruis applied to a local maximum and,critical point are structurally stable 1

The implication of this result is tlis an unbounded operator, the dyruchange in the Krein signature of thecondition for structural instability. 1the signature changes, however. Oncan give birth to unstable modes.

4

0.80.6

(a)

(b)

0.2 0.4

H[fo+xJ

-1 0v

-2-3

0.8

0.6

0.4

0.2><+~

-0.2

-0.4

-0.6

-0.8

-1-5 - 4

0.8

0.6

0.4

0.2><I~I -0.2

- 0.4

-0.6

- 0.8

-1-0.4 -0.2

The reason that all homogeneous equilibria were structurally unstable in theprevious section was because small perturbations could create regions of differentsignature near critical points of fa. In fact, the Penrose criterion requires that fa has aminimum in order for there to be an unstable mode. The perturbation X that was usedto destabilize fa created a distribution function with derivative f~ + X that had a localminimum at what was previously a local maximum of fa. However, dynamicallY

Dynamical accessibility also elmmodes. Dynamically accessible peruSince f~ (uc ) changing sign at soimpossible for a dynamically accesPoint mode and otherwise only a c(unstable. This is consistent with tsignature of the continuous spectnuare both positive.

,Because [q ,p] = 1 and p(p) is not. If the spatial domain is finite, thisifinite spatial domains, this is not a

0.6 0.8

ion, f~ +X. b) The Hilbert


accessible perturbations cannot change level set topology and, consequently, thenumber of critical points of fa. Indeed, if (q ,p) is an area preserving diffeomorphismand p is a homogeneous, i.e. a function of p alone, then the critical points of fa(p)are the points p-I (Pc), where Pc is a critical point of fa(p). By the chain rule, thesecritical points will always be the same type as the corresponding critical point of fa the map Pmust be monotonically increasing in order for it to be invertible.

One implication is that the perturbation X is not dynamically accessible when itis applied to a local maximum and, consequently, all equilibria fa with only a singlecritical point are structurally stable under dynamically accessible perturbations.

If Pc is a non-degenerate critical point of fa such that f~ (Pc) < 0, then theprevious obstruction to the application of X using a dynamically accessibleperturbation does not apply. In [HAG 11b], it is shown that there is a rearrangementp such that fa(p) fa(p) + J!..oodp' X(p' - Pc) or thatdfa(p)/dp = f6(p) + X(p - Pc). Such a rearrangement can be constructed as long asthe parameters defining X, the numbers h.d ,e, are chosen such thatf6(p) +X(p - Pc) has the same critical points as f6(p). The construction uses Morsetheory to find -a p so that fa(p) = fa(p) + J X + O((p - Pc)3), where O((p - Pc?)has compact support and is smaller than fa(p) - f6'(pc)(p - Pc)2 /2.

These ideas lead to the following Krein-like theorem for dynamically accessibleperturbations in the WI ,I norm:

THEOREM 13.2.- Let fa be a stable equilibrium distribution function for the Vlasovequation on an infinite spatial domain. Then, fa is structurally stable underdynamically accessible perturbations in WI ,I (IR) if there is only one solution off6(p) = 0. If there are multiple solutions, fa is structurally unstable and the unstablemodes come from the zeros of f6 that satisfy f6' (p) < 0.

The implication of this result is that in a Banach space where the Hilbert transformis an unbounded operator, the dynamical accessibility condition makes it so that achange in the KreIn signature of the continuous spectrum is a necessary and sufficientcondition for structural instability. The bifurcations do not occur at all points wherethe signature changes, however. Only those that represent valleys of the distributioncan give birth to unstable modes.

vere structurally unstable in thecould create regions of differentrse criterion requires that fa has aThe perturbation X that was usedderivative f6 +X that had a local

rm of fa. However, dynamically

Dynamical accessibility also clarifies bifurcations to instability of inflection pointmodes. Dynamically accessible perturbations cannot eliminate inflection points of fa.Since f6(uc ) changing sign at some point Uc is necessary for instability, it isimpossible for a dynamically accessible perturbation of an fa that has an inflectionpoint mode and otherwise only a continuous spectrum with positive signature to beunstable. This is consistent with the fact that there exists a frame in which thesignature of the continuous spectrum and the signature of the inflection point modeare both positive.


13.4. Canonical infinite-dimensional case

There have been some works on structural stability of infinite-dimensionalHamiltonian systems. The first of these results is again due to Krein and recorded inhis book with DaleckiI [DAL 70] on ordinary differential equations on Banachspaces. They considered the simplest possible infinite-dimensional Hamiltoniansystems; canonical equations with bounded time evolution operators on Hilbertspaces. They defined signature in terms of positive and negative splittings of thecanonical symplectic 2-form (Lagrange bracket) on the Hilbert space, the resultingcondition derived in this case is the following: if the part of the spectrumcorresponding to the positive space overlaps with the part of the spectrumcorresponding to the negative case, then there is an infinitesimal perturbation thatcauses the system to become unstable. This result applies when there is a continuousspectrum as well as a discrete spectrum, and is a direct generalization of Krein'sfinite-dimensional theorem. The splitting of the spectrum into positive and negativesignature subspaces can be converted into an equivalent splitting in terms of positiveand negative energy, though delicacy is again required when the spectrum iscontinuous. In these cases, we look at whether the Hamiltonian operator is positive ornegative definite on the spectral projections onto the targeted parts of the spectrum.The slightly different definition of signature is useful when the Hamiltonianfunctional is allowed to depend on the time t, which was also studied by KreIn, butotherwise is equivalent to our definition.

The situation is more complicanwas considered by Grillakis [ORI ~

of travelling waves in the nonlineasystems. He was also interested in dof negative eigenvalues, a problem[CHU 10, KAP 04]. In the case whethe continuous spectrum (which halwas able to prove structural instabil:under the assumption of relatively bstructural stability. It should be notecspectrum is different than in the c.media field theories that exhibit CHIaction of a derivative operator on athan a multiplication operator. In thenonlinear evolution and saturation (similar equations is described by sorinteresting to see if there is an analoat least in some sense. This would bof continuous spectra are related.

13.4.1. Negative energy oscillator (

where J is an anti symmetric unitary operator which without loss of generality can be

assumed to be J = (~ ~I) , and 21 is the self-adjoint operator associated with some

sesquilinear form H[·,.J.Equation [13.21] is a Hamiltonian system with Hamiltonianfunctional H[! ,!J. KreIn said this system was strongly stable (structurally stable inour terminology) if there is some 8 > 0 such that for alll~l - ~I < 8 , the spectrum ofJ~1 is contained in the imaginary axis. KreIn was able to prove that the system wasstrongly stable if and only if the phase space !Jll splits into two subspaces, !Jll+ and ~- ,

each invariant under the time evolution operator J~ such that J is positive on !Jll+ andnegative on !Jll_. This is equivalent to the Hamiltonian operator ~ being positive on!Jll+ and negative on !Jll_ , which means that the system is structurally stable as long aspositive energy parts of the spectrum are disjoint from the negative energy parts of thespectrum. No reference is made to the type of spectrum of the operator J~, and thesign of the operator ~ on the eigenspace corresponding to some part of the spectrumdefines the signature of that part of the spectrum.

In particular, KreIn examined canonical equations of the form

it =J~!, [13.21]

An illustrative example of the Ca negative energy version of the Callnoncanonical equations considered j

spectrum arises from a multiplicatsimple model of a discrete mode enintroduce dissipation into quantunprocess of phase mixing, essentiallyquantum mechanics. (Landau damjwhich leads to highly oscillatorydetermined by the Riemann-Lebesgof the discrete oscillator, we altgyroscopically stabilized system iet at. [BLO 04]). This results in strthe amplitude of the coupling term.of the Nyquist method, resulting in ;:

stability of infinite-dimensionaligain due to KreIn and recorded indifferential equations on Banachinfinite-dimensional Hamiltonian

.e evolution operators on Hilbertive and negative splittings of them the Hilbert space, the resultingg: if the part of the spectrumwith the part of the spectruman infinitesimal perturbation that

applies when there is a continuousa direct generalization of Krein'slectrum into positive and negative'alent splitting in terms of positiverequired when the spectrum is

1:amiltonian operator is positive orhe targeted parts of the spectrum.~ useful when the Hamiltonianch was also studied by KreIn, but

ns of the form

[13.21]

l without loss of generality can be

lint operator associated with some

iltonian system with Hamiltonianngly stable (structurally stable in.alll~l - ~I < 8, the spectrum ofable to prove that the system was.into two subspaces, ~+ and ~_,such that J is positive on ~+ andiian operator ~ being positive on.m is structurally stable as long asm the negative energy parts of thetrum of the operator J~, and theling to some part of the spectrum


The situation is more complicated when ~ is allowed to be unbounded. This casewas considered by Grillakis [GRI90], who was interested in studying the stabilityof travelling waves in the nonlinear Schrodinger (NLS) equation and other similarsystems. He was also interested in developing a technique for determining the numberof negative eigenvalues, a problem subsequently discussed by a number of authors[CHU 10, KAP 04]. In the case where there was a negative energy mode embedded inthe continuous spectrum (which had positive signature in those examples), Grillakiswas able to prove structural instability. In the case where all signatures were positive,under the assumption of relatively bounded perturbations, Grillakis was able to provestructural stability. It should be noted that in the NLS case, the nature of the continuousspectrum is different than in the case of Vlasov-Poisson and the other continuousmedia field theories that exhibit CHH bifurcations. In the NLS equation, it is due to theaction of a derivative operator on a function space over an unbounded domain ratherthan a multiplication operator. In the last section of this chapter, we will argue that thenonlinear evolution and saturation of the resulting instability of Vlasov-Poisson andsimilar equations is described by something called the single-wave model. It would beinteresting to see if there is an analog of the single wave model for systems like NLS,at least in some sense. This would be related to the greater issue of how the two typesof continuous spectra are related.

13.4.1. Negative energy oscillator coupled to a heat bath

An illustrative example of the CHH bifurcation in the canonical case comes froma negative energy version of the Caldeira-Leggett model. This case is like that for thenoncanonical equations considered in the bulk of this chapter because the continuousspectrum arises from a multiplication operator. The Caldeira-Leggett model is asimple model of a discrete mode embedded into a continuous spectrum. It is used tointroduce dissipation into quantum mechanics [CAL 81, HAG lla] through theprocess of phase mixing, essentially realizing the phenomenon of Landau damping inquantum mechanics. (Landau damping is a symptom of the continuous spectrum,which leads to highly oscillatory solutions whose moments decay with time asdetermined by the Riemann-Lebesgue lemma). By flipping the sign of the signatureof the discrete oscillator, we alter the Caldeira-Leggett model to describe agyroscopically stabilized system interacting with a heat bath (see also Blochet al. [BLO 04]). This results in structural instability, where the small parameter isthe amplitude of the coupling term. We demonstrate this result through an adaptationof the Nyquist method, resulting in a Penrose-like criterion for stability.


The Hamiltonian for this system is

H[Q ,P,q(x),p(x)] Q (2 2) 1 t" 2-2 Q +P +"2 Jo dxx(q(x) + p(xf)

+Q l<Xldx f (x)q(x) . [13.22]

If we divide both sides by ro2+ f)

from the upper half plane, we get theon the real axis:

ro2 _ Q2e(ro) = -ro-2 -+-Q-2

If f(x) = 0, the Hamiltonian describes a system consisting of a single harmonicoscillator with negative energy and a continuous bath of oscillators with positiveenergy, where (q(x),P(x)) are coordinates for the bath and here (Q,P) the singleharmonic oscillator. Solutions are stable and consist of independent oscillations ofthe discrete oscillators and the continuum. If the discrete oscillator has positiveenergy, and we activate the coupling to the continuum, then because the spectrum isalways of positive signature, we will still have stable solutions. In the negativesignature case, we expect the opposite. This can be seen by an argument that isanalogous to the Penrose criterion in the Vlasov equation.

The equations of motion are

Using the argument principle, weplane is equal to the winding numbeiminus the number of poles. Since thof zeros is the winding number plusnegative; for x < 0, it is positive. Faof this contour will be 1, and there vupper half plane, see Figure. 13.6 fan interaction between the continuorsignature, just like in the CHH bifurc

O.3r---------.----

dQ-=-QPdt

aq(x) = xp(x)at

dP ( <Xldt = QQ- Jo dxf(x)q(x)

ap(x)-at = -xq(x) - Qf(x) ,

[13.23]

[13.24]

o

o

which have the dispersion relation

[13.25]

Here, we use partial fractions to write the integral on the right hand side of [13.25]in terms of the Cauchy integral of the anti-symmetric extension of f(x), denoted byf-(x),

-oAL...-.----------l....----0.4

Figure 13.6. Nyquist plot for tlenergy harmonic oscillator at ,

f(x

[13.26] 13.5. Commentary: degeneracy an

We have given arguments above i

HH except the continuous spectrum 'pairs in the discrete case. A telltale 'role is the presence of an imaginary ;real frequencies, as is the case, e.g. i


If we divide both sides by 002 +.Q.2 and take the limit as 00 approaches the real axisfrom the upper half plane, we get the following expression for the dispersion relationon the real axis: .1 (00

2io dxxlq(x)2+p(x)2)

dxf(x)q(x) . [13.22]

002 _.Q.2

£ ( (0) = -00--0-2 -+-.Q.-2 [13.27]

:em consisting of a single harmoniclS bath of oscillators with positivehe bath and here (Q,P) the singlensist of independent oscillations ofthe discrete oscillator has positivenuum, then because the spectrum ise stable solutions. In the negativem be seen by an argument that isquation.

Using the argument principle, we find that the number of zeros in the upper halfplane is equal to the winding number of the image of the real line under this mappingminus the number of poles. Since there is a single pole, where 00 = i.Q. , the numberof zeros is the winding number plus 1. For x > 0, the imaginary part of the image isnegative; for x < 0, it is positive. For generic f (not too large), the winding numberof this contour will be 1, and there will be two zeros of the dispersion relation in theupper half plane, see Figure. 13.6 for such an example. These zeros emerge due toan interaction between the continuous spectrum and the discrete mode with oppositesignature, just like in the CHH bifurcations that we have discussed so far.

O.3r------- -,..---------------------,

f(x)q(x )

-Qf(x) ,

[13.23]

[13.24]

o

o

[13.25]

ral on the right hand side of [)13.25]etric extension of f(x), denoted by

[13.26]

-oA~-------L...----------------------J

-0.4

Figure 13.6. Nyquist plot for the Caldeira-Leggett model with a negativeenergy harmonic oscillator at frequency .Q = 1.0 and coupling function

f(x)2 = .4xe-·25x2

13.5. Commentary: degeneracy and nonlinearity

We have given arguments above and in Chapter 12 that the CHH is like the usualHH except the continuous spectrum plays the role of one of the colliding eigenvaluepairs in the discrete case. A telltale sign that a continuous spectrum is playing thisrole is the presence of an imaginary part in the dispersion relation when evaluated atreal frequencies , as is the case, e.g. for the Vlasov-Poisson system where on the real


where the function G is real. This isand imaginary parts

where u = OJ/ k = UR + iuj. Then,its argument P and splitting the insymmetric parts yields

e(k ,OJ)

u axis e(k,u) := 1 - nk? £[f~](u) ± iJrk- 2f~(u). The two signs occur because thereal axis is a branch cut, which is known to be a consequence of continuous spectrain systems of this type. Observe that the same occurs for the Caldeira-Leggett modelin [13.27]. After collision, the number of discrete eigenvalues that emerge can becounted in a straightforward way. For example, consider the Penrose plot ofFigure 13.3.1 that depicts a k =I- 0 bifurcation at criticality. If the fo(p) used forFigure 13.3.1 is replaced by fT](p) , a one-parameter perturbation that matches fo(p)at 11 = 0, then when instability sets in, the point of tangency will move with 11 so thatthere are two intersections of the real axis giving rise to a winding number of unity,which signals the instability. Generically, this will give a complex eigenvalue whereOJ = OJR + iy, with both real and imaginary parts of OJ depending on the bifurcationparameter 11. A similar Penrose argument reveals that there is also a root in the lowerhalf plane, bringing our eigenvalue count to two, with OJ = OJR - iy corresponding todecay. In these plots, k is assumed to be fixed, but associated with a given kE N is acanonical pair, (.!2k, 9 k), which can be traced back to t;k and t;-k. Here, each kENlabels a degree of freedom, which has two associated eigenvalues: a mode or degreeof freedom, determined by its wave number, has two dimensions, corresponding toamplitude and phase . Replacing k with -k in e gives the remaining two eigenvalues,OJ = -OJR ± iy. Thus, the CHH is a bifurcation to a quartet, OJ = ±OJR ± iy, and afterbifurcation, the structure is identical to that of the ordinary HH bifurcation.

Tractability often arises in problems because of assumptions of symmetry, e.g.the homogeneity of the equilibrium fo simplifies the Vlasov problem and thesymmetry in the Jeans problem of section 12.2.2.3 of Chapter 12 allowed an explicitsolution of the dispersion relation [12.25] . Thus, the question arises, what happens ifwe symmetrize the k =I- 0 CHH bifurcation discussed above? If fo(p) = fo(-p), withthe upper portion of Figure 13.3.1 unchanged, then we obtain a plot that is reflectionsymmetric about the £ -axis with two osculating points. Under ~ parameter changeto instability, both curves must cross , and using the ray counting procedure discussedin section 13.3, this causes the winding number to jump by 2. Thus , for symmetric fowith k =I- 0, bifurcating eigenvalues occur in pairs , and after bifurcation, we have anoctet, characteristic of a degenerate CHH.

Next, consider the k = 0 bifurcation with the imposed symmetry fT] (p) = fT] (-~)for all control parameter values 11 E IR20 with criticality at 11 = 0 as depicted III

Figure 13.3. Because of the symmetry, fT] (0) = 0 for all 11 near 11 = O. Thebifurcation can be instigated either by fixing k and varying 11 or by setting 11 to avalue for which the crossing of Figure 13.3 becomes negative and then varying kuntil e = O. Either way, it follows that with the imposition of this symmetry, thesolution of the dispersion relation, e = 0, must have the following form:

[13.28]

This expression must vanish whthe integral equals zero. If we assuthe integral of [13.30] vanishes. Innot expect the integral to vanish; thedoes not reference this integral in (relation at criticality for such only (mode. Therefore, at the bifurcation

Note that the case of the degeneargument due to the potential vanisland UR to be non-zero. From such a ~

solutions in the upper half plane. AI:the sign of the interaction is reversethe Maxwellian distribution and the

From the above discussion abo11 =0, G(k,O) = 0 implies discreteG(k,11 > 0) < 0 implies two pungrowth and decay. In fact, the situ[12.22] for the multifluid exameigenvalues as above, we see that a


where the function G is real. This is seen by separating the dispersion relation into realand imaginary parts

where U = ill / k = UR + iuj. Then, with the assumption that f~ is antisymmetric inits argument p and splitting the imaginary part of [13.29] into symmetric and antisymmetric parts yields

t). The two signs occur because the. consequence of continuous spectracurs for the Caldeira-Leggett modelete eigenvalues that emerge can beole, consider the Penrose plot ofat criticality. If the fo(p) used foreter perturbation that matches fo(p)If tangency will move with 11 so that~ rise to a winding number of unity,11 give a complex eigenvalue where, of ill depending on the bifurcationthat there is also a root in the lowerwith ill = illR - iy corresponding toIt associated with a given kE N is atck to ~k and ~-k' Here, each kE Nated eigenvalues: a mode or degreei two dimensions, corresponding toives the remaining two eigenvalues,I a quartet, ill = ±illR ± iy, and afterordinary HH bifurcation.

e(k, ill)

: of assumptions of symmetry, e.g.ties the Vlasov problem and the3 of Chapter 12 allowed an explicitthe question arises, what happens ifsed above? If fo(p) = fo(-p), withm we obtain a plot that is reflection~ points. Under a parameter changele ray counting procedure discussedI jump by 2. Thus, for symmetric fo;, and after bifurcation, we have an

nposed symmetry f Ti (p) = f Ti ( - p)criticality at 11 = 0 as depicted in= 0 for all 11 near 11 = O. Thend varying 11 or by setting 11 to aomes negative and then varying k: imposition of this symmetry, there the following form:

[13.28]

This expression must vanish when u is a root, which implies that u/ = 0, UR = 0 orthe integral equals zero. If we assume that ut is non-zero, then either UR vanishes orthe integral of [13.30] vanishes. In general, even with the assumed symmetry, we donot expect the integral to vanish; the condition for the existence of an embedded modedoes not reference this integral in any way, and the imaginary part of the dispersionrelation at criticality for such only depends on the value of f Ti at the frequency of themode. Therefore, at the bifurcation point, this integral does not appear.

Note that the case of the degenerate octet discussed above is not forbidden by thisargument due to the potential vanishing of the integral , which would allow for both u/

and UR to be non-zero. From such a state, further variation of f Ti will lead to a branch ofsolutions in the upper half plane. Also note that for the Vlasov-Jeans instability, wherethe sign of the interaction is reversed, the integral [13.30] has a positive integrand forthe Maxwellian distribution and therefore cannot vanish.

From the above discussion about symmetry, it is clear that at criticality, say at11 = 0, G(k,O) = 0 implies discrete zero frequency eigenvalues, while as 11 increases,G(k,11 > 0) < 0 implies two pure imaginary eigenvalues , in?icati~g expo~ential

growth and decay. In fact, the situation is precisely like the dispersion relation of[12.22] for the multifluid example of Chapter 12. Upon properly countingeigenvalues as above, we see that after the bifurcation there are in fact two growing


and two decaying eigenvalues. We note that an attempt to use the usual marginalityrelations for determining the eigenvalues,

at OJR = 0, will be indeterminant because both the numerator and denominator of rvanish. As we have seen in Chapter 12 such degenerate steady-state (SS) bifurcationshappen in finite systems when symmetry is imposed. We call any SS bifurcation in thepresence of a continuous spectrum, a CSS bifurcation.

e/(k,OJR)aeRIaOJR '

[13.31]

responsible for Landau damping oranalyses to fail because of singuHowever, in [BAL 13], it was showthe single-wave model emerges fro iChapter 12. An essential ingrediHamiltonian systems have a contarising not from an infinite spatial (in phase space (e.g. the wave-partlevels in fluid mechanics). Thus, ththe CHH and CSS bifurcations.

In particular, the SW model dewhich have been rediscovered in dilmodel features the so-called trappithe cats-eye or phase space hole strpatterns. An example of this is shopattern and temporal fate of the simodel also gives a description of ncan be arrested by nonlinearity. Anthe scope of the present contributioform that aligns with the CHH bifuwith the CSS bifurcation. We refe:further details .

Figure 13.7. Evolution of the sirphase space hole pattern. b) Bel

If we break the symmetry, then generically as T] increases, the k = 0 bifurcation isa CHH bifurcation. For this case generally, f~ does not vanish and equations [13.31]apply. Counting eigenvalues gives the CHH quartet. One might be fooled intothinking a change of frames, a Doppler shift, would make the symmetric andnon-symmetric k = 0 bifurcations identical, but this is not the case. Galilean frameshifting the degenerate CSS, say by a speed v", replaces [13.28] by a dispersionrelation of the form (OJ - kv*)2 = G(k,T]); thus, unlike the non-symmetric case, thereal parts of the frequencies do not depend on T].

A goal of linearized theories is to predict weakly nonlinear behavior. Indeed,bifurcation theory in dissipative systems has achieved great success in this regard. Inparticular, for finite-dimensional systems rigorous center manifold theorems allow usto reliably track bifurcated solutions into the nonlinear regime and, in someinstances, obtain saturated values. For infinite-dimensional systems, various normalforms, such as the Ginzburg-Landau equation, adequately describe pattern formationdue to the appearance of a single mode of instability in a wide variety of dissipativeproblems. In Hamiltonian systems, the situation is more complicated; the lack ofdissipation creates a greater challenge because dimensional reduction is not soaccommodating. However, for finite-dimensional Hamiltonian systems, there is along history of perturbation/averaging techniques for near-integrable systems,systems with adiabatic invariants, etc. Techniques that may lead to nonlinear normalforms. Similarly, techniques have been developed for infinite-dimensionalHamiltonian systems, particularly in the context of single field 1+1 models. However,the combination of nonlinearity together with the type of continuous spectrumdiscussed here and in Chapter 12 provide a distinctively more difficult challenge.

This challenge is met by the single-wave (SW) model, an infinite-dimensionalHamiltonian system that describes the behavior near threshold and subsequentnonlinear evolution of a discrete mode that emerges from the continuous spectrum.The SW model was originally derived in plasma physics, then (re)discovered invarious fields of inquiry, ranging from fluid mechanics, galactic dynamics a~dcondensed matter physics. The presence of the continuous spectrum, which IS

(a)

20

10

>, 0

-10

-20

o 2 4

o

ttempt to use the usual marginality

[13.31]

e numerator and denominator of rerate steady-state (SS) bifurcationsd. We call any SS bifurcation in theion.

1 increases, the k = 0 bifurcation is~s not vanish and equations [13.31]iartet. One might be fooled intowould make the symmetric and

his is not the case. Galilean framereplaces [13.28] by a dispersion

mlike the non-symmetric case, the

eakly nonlinear behavior. Indeed,ved great success in this regard. Incenter manifold theorems allow usnonlinear regime and, in some

rensional systems, various normalquately describe pattern formationity in a wide variety of dissipativeis more complicated; the lack ofdimensional reduction is not soHamiltonian systems, there is a

res for near-integrable systems,that may lead to nonlinear normal'eloped for infinite-dimensionalsingle field 1+1 models. However,he type of continuous spectrumively more difficult challenge.

V) model, an infinite-dimensional. near threshold and subsequentes from the continuous spectrum.1 physics, then (re)discovered inechanics, galactic dynamics and

continuous spectrum, which is


responsible for Landau damping on the linear level, causes conventional perturbationanalyses to fail because of singularities that occur at all orders of perturbation.However, in [BAL 13], it was shown by a suitable matched asymptotic analysis, howthe single-wave model emerges from the large class of2+1 theories of section 12.3 ofChapter 12. An essential ingredient for this asymptotic reduction is that theseHamiltonian systems have a continuous spectrum in the linear stability problemarising not from an infinite spatial domain but from singular resonances along curvesin phase space (e.g. the wave-particle resonances in the plasma problem or critical'levels in fluid mechanics). Thus, the SW model describes nonlinear consequences ofthe CHH and CSS bifurcations.

In particular, the SW model describes a range of universal phenomena, some ofwhich have been rediscovered in different contexts. For a bifurcation to instability, themodel features the so-called trapping scaling dictating the saturation amplitude andthe cats-eye or phase space hole structures that characterize the resulting phase-spacepatterns. An example of this is shown in Figure 13.7, which depicts the phase spacepattern and temporal fate of the singe-wave (bifurcated mode) amplitude. The SWmodel also gives a description of nonlinear Landau damping, i.e. how such dampingcan be arrested by nonlinearity. An in-depth description of the SW model is beyondthe scope of the present contribution, but we comment that in addition to the normalform that aligns with the CHH bifurcation, there is also a degenerate from associatedwith the CSS bifurcation. We refer the reader to [BAL 13] (notably section V) forfurther details.

x

Figure 13.7. Evolution of the single-wave model [BAL 13]. a) Plot ofa typicalphase space hole pattern. b) Behavior ofmode amplitude as a function of time


13.6. Summary and conclusions

We presented a mathematical account of CHH bifurcations in 2+I Hamiltoniancontinuous media field theories. We presented a mathematical framework in whichwe describe the structural stability of equilibria of Hamiltonian systems, whose mostimportant ingredient was a method for attaching signature to the continuousspectrum. We presented an application of this framework to the Vlasov-Poissonequation, demonstrating that the two-stream instability can be interpreted as apositive energy mode interacting with a negative energy continuous spectrum, andthat all equilibria are structurally unstable in Banach spaces that are not strongenough to prevent infinitesimal perturbations from altering the signature of thecontinuous spectrum. If we restrict to dynamically accessible perturbations, which byconstruction cannot effect the signature, then only those equilibria with both positiveand negative signatures are structurally unstable.

Finally, in this chapter, we examined the difference between canonical andnoncanonical Hamiltonian systems and also the idea that the single-wave model is anormal (degenerate) form for the CHH (CSS) bifurcation that describes its nonlinearevolution. These processes underlie phase space pattern formation in the 2+I theoriesthat we are interested in, and explaining these patterns and the structure of the phasespace of these systems provides strong motivation for this work and further research.

13.7. Acknowledgments

Hospitality of the GFD Summer Program, held at the Woods Hole OceanographicInstitution, is greatly appreciated. G.!. Hagstrom and PJ. Morrison were supported byUSDOE grant nos. DE-FG02-ER53223 and DE-FG02-04ER54742, respectively.

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fifference between canonical andlea that the single-wave model is aircation that describes its nonlinearattem formation in the 2+1 theoriesterns and the structure of the phasefor this work and further research.

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Energy StabilitFluid-Kin

In plasma physics, a hylmagnetohydrodynamics (MHD) I=

Vlasov kinetic theory part that de:most hybrid models in the plasinvestigates a recent Hamiltoniancorresponding Hamiltonian strucnThen, the energy-Casimir methcconditions, which imply a stable s1

14.1. Introduction

The discipline of plasma physstability problems. Depending onfluid models or kinetic theories wiplasma theory has all of the possildisciplines and, although a great dthere remain many open mathemat

When dissipative terms are Sf

plasma systems are infinite-dimenstability and spectral problems ofthese operators are non-Hermitian,

Chapter written by Philip J. MORRISO

Documents

SpectralAnalysis, Stability andBifurcations Systems ...morrison/13BIRS_morrison_CHHII.pdf · systems. In this chapter, we describe some mathematical aspects of the CHH continuing